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Abstract

Many combinatorial identities can be formulated in terms of
q-hypergeometric sums, for instance, the celebrated Rogers-Ramanujan
identities from additive number theory. Identities of this type can be
constructed iteratively from simpler ones, i.e., by proceeding along
Bailey chains. Another construction mechanism, different from this
classical one, arises within the context of q-WZ-theory. For
instance, as a by-product of computer proofs, one automatically
obtains the so-called ``dual'' identities. The talk gives a short
tutorial introduction and discusses various relations between
these concepts.

The talk consists of four parts. The first part is an introduction to
Gaussian polynomials. The second part is a brief account of
q-hypergeometric WZ theory. The parts that follow are variations
on this theme.

1 Gaussian polynomials

Let p(m,n;k) denote the number of partitions of k in at most
m parts, each part £n . Clearly,

Using the standard notation (a;q)k=(1-a)(1-aq)···(1-aqk-1),
k = 1,2,... and (a;q)k=1/(a;q)-k for k<0, we get a closed
form representation
of the Gaussian polynomials Gm,n :

éë

m+n

n

ùû

:= Gm,n(q)
=

(1-qm+n) ··· (1-qm+1)

(1-qn) ··· (1-q)

=

(q;q)m+n

(q;q)m (q;q)n

.
(3)

2 Some facts about q-hypergeometric WZ theory

Definition 1
A sequence (tk) is hypergeometric if the
ratio of two consecutive terms is a rational function of the summation
index k: tk+1/tk=P(k)/Q(k), where P and Q are
polynomials in k.

Definition 2
A sequence (tk) is q-hypergeometric if the
ratio of two consecutive terms is a rational function of qk:
tk+1/tk = P(qk)/Q(qk),
where P and Q are polynomials in qk. (Note that q should be
contained in the coefficient field which should be of characteristic 0.)

2.1 From Gosper to Zeilberger

Gosper's algorithm for indefinite hypergeometric
summation [3] is
given as input a hypergeometric sequence (fk) . This algorithm finds
a hypergeometric sequence (gk) such that fk = gk+1 - gk
(then gk is the product of fk and a rational function in k).
From there by telescoping one gets:

n

å

k=0

fk = gn+1 - g0.

Zeilberger's algorithm computes definite hypergeometric sums.
Given a proper hypergeometric sequence (Fn,k)
(with finite support in k), this algorithm finds a hypergeometric
sequence (Sk) such that

å

k

Fn,k = Sn.

The idea is to use an extension of Gosper's algorithm in order to find
polynomials aj(n) and a proper hypergeometric term (Gn,k)
such that

J

å

j=0

aj(n) Fn+j,k = Gn,k+1 - Gn,k.

Then Gn,k is necessarily of the form R(n,k) Fn,k where R is a rational function called the ``certificate''.
Summing this equality yields the desired recurrence on Sn:

J

å

j=0

aj(n) Sn+j = 0.

This algorithm has been implemented in Mathematica by P. Paule and M. Schorn:

Both these algorithms extend to the q-case, a corresponding
implementation in Mathematica is due to A. Riese.

2.2 Example

A variation of a q-analogue of Gosper's and Zeilberger's algorithms can serve
in finding q-analogues of binomial identities. For instance, in order to
derive a q-analogue of the binomial theorem, the question is to find
a, bÎZ such that the following sequence
satisfies a recurrence of order one:

By using this in the form
a2(n)/Sn+1 = - a1(n)/Sn , we
rewrite the relation above as:

a1(n)

Sn

fn+1,k

Sn+1

+

a2(n)

Sn+1

fn,k

Sn

=

gn,k+1 - gn,k

Sn+1Sn

.

Defining Fn,k = fn,k/Sn ,
and Gn,k = gn,k/(a1(n)Sn+1) , we arrive at:

Fn+1,k - Fn,k = Gn,k+1 - Gn,k.
(4)

This gives rise to the following definition:

Definition 3
A pair of sequences (F,G) that satisfy (4) is
called a ``WZ pair''.

Note that given such a WZ-pair (F having finite support),
from (4) follows that
åkFn+1,k - åkFn,k = 0 ,
which means that the corresponding sum sequence Sn := åkF(n,k)
is a constant.

3 A Fibonacci q-analogue

The well-known Fibonacci numbers are defined by F0 = 1, F1 = 1
and Fn+2 = Fn+1 + Fn . Does there exist a q-analogue of these
numbers?
In order to follow the strategy explained above (example 2.2),
we take as a starting point the following well-known hypergeometric sum:

Fn =

n

å

k=0

æè

n-k

k

öø

.

For the (a, b)-Ansatz, we take:

Fn(q) =

n

å

k=0

éë

n-k

k

ùû

q

a

æè

k

2

öø

+ bk

,

and we want to determine a, bÎZ such that (Fn(q))
satisfies a linear recurrence of order 2. Riese's implementation delivers as
candidates: aÎ {1,2,3} and bÎZ , but only the
choice a = 2 is successful. This means, only for a = 2 ,
the q-analogue of Zeilberger's algorithm delivers ("bÎZ ) a recurrence of order 2, namely:

Fn+2(q) = Fn+1(q) + q

n+b

Fn(q).
(5)

Let us fix (for instance) b = 1 , and we obtain for this choice:

Fn(q) =

n

å

k=0

éë

n-k

k

ùû

q

k2

.

In the limit n®¥ :

F

¥

(q) =

¥

å

k=0

q

k2

(q;q)k

= 1 +

¥

å

n=1

bnqn
=

¥

Õ

k=1

1

(1-q5n+1)(1-q5n+4)

,
(6)

where bn is the number of partitions of n into parts with minimal
difference two and the right-hand side is one of the celebrated
Rogers-Ramanujan identities [1].

Starting from (5), it is also possible to
conjecture and then prove (by q-Zeilberger) the following identity
due to I. Schur

F2n(q)=

å

k

qk(10k+1)

éë

2n

n-5k

ùû

-

å

k

q(2k-1)(5k-2)

éë

2n

n-5k+2

ùû

.

4 The Bailey chain approach

Proposition 1

n

å

j=0

q

j2-k2

(q;q)n-j

(q;q)n-k (q;q)n+k

(q;q)j-k (q;q)j+k

= 1.
(7)

Proof. Denote by fn,j the summand. Riese's
implementation yields:

fn,j - fn-1,j = gn,j - gn,j-1,
where gn,j
=

qk+n (qj - qn)

(qn - qk)(1-qk+n)

fn,j.

This (q)WZ-pair implies that the sum over fn,j is constant.
That this constant is 1 follows from instance from the evaluation
for n=k [1, 4].
This identity is a special case of a q-hypergeometric formula
that can be proved combinatorially as explained in [4].

Multiplying (7) by an
arbitrary sequence (ck), we obtain the following special case of
``Bailey's Lemma'':

å

k

ck

(q;q)n-k (q;q)n+k

=

å

j³ 0

q

j2

(q;q)n-j

å

k

ckq

-k2

(q;q)j-k (q;q)j+k

.
(8)

Definition 4 Two sequences ((a)kÎZ, (b)nÎN)
are called a Bailey-pair when

References

Andrews (George E.). --
q-series: their development and application in analysis,
number theory, combinatorics, physics, and computer algebra. --
Published for the Conference Board of the Mathematical Sciences,
Washington, D.C., 1986, CBMS Regional Conference Series
in Mathematics, vol. 66, xii+130p.

Paule (Peter) and Riese (Axel). --
A Mathematica q-analogue of Zeilberger's algorithm based on an
algebraically motivated approach to q-hypergeometric telescoping. In Special functions, q-series and related topics, pp. 179--210. --
American Mathematical Society, Providence, RI, 1997.
Proceedings of a workshop held in Toronto, ON, June 1995.